Preface. About the Editors. Part I: Quantum Computing. 1. Quantum computing with qubits S.L. Braunstein, A.K. Pati. 2. Quantum computation over continuous variables S. Lloyd, S.L. Braunstein. 3. Error correction for continuous quantum variables S.L. Braunstein. 4. Deutsch-Jozsa algorithm for continuous variables A.K. Pati, S.L. Braunstein. 5. Hybrid quantum computing S. Lloyd. 6. Efficient classical simulation of continuous variable quantum information processes S.D. Bartlett, B.C. Sanders, S.L. Braunstein, K. Nemoto. Part II: Quantum Entanglement. 7. Introduction to entanglement-based protocols S.L. Braunstein, A.K. Pati. 8. Teleportation of continuous uantum variables S.L. Braunstein, H.J. Kimble. 9. Experimental realization of continuous variable teleportation A. Furusawa, H.J. Kimble. 10. Dense coding for continuous variables S.L. Braunstein, H.J. Kimble. 11. Multipartite Greenberger-Horne-Zeilinger paradoxes for continuous variables S. Massar, S. Pironio. 12. Multipartite entanglement for continuous variables P. van Loock, S.L. Braunstein. 13. Inseparability criterion for continuous variable systems Lu-Ming Duan, G. Giedke, J.I. Cirac, P. Zoller. 14. Separability criterion for Gaussian states R. Simon. 15. Distillability and entanglement purification for Gaussian states G. Giedke, Lu-Ming Duan, J.I. Cirac, P. Zoller. 16. Entanglement purification via entanglement swapping S. Parke, S. Bose, M.B. Plenio. 17. Bound entanglement for continuous variables is a rare phenomenon P. Horodecki, J.I. Cirac, M. Lewenstein. Part III: Continuous Variable Optical-Atomic Interfacing. 18. Atomic continuous variable processing and light-atoms quantum interface A. Kuzmich, E.S. Polzik. Part IV: Limits on Quantum Information and Cryptography. 19. Limitations on discrete quantum information and cryptography S.L. Braunstein, A.K. Pati. 20. Quantum cloning with continuous variables N.J. Cerf. 21. Quantum key distribution with continuous variables in optics T.C. Ralph. 22. Secure quantum key distribution using squeezed states D. Gottesman, J. Preskill. 23. Experimental demonstration of dense coding and quantum cryptography with continuous variables Kunchi Peng, Qing Pan, Jing Zhang, Changde Xie. 24. Quantum solitons in optical fibres: basic requisites for experimental quantum communication G. Leuchs, Ch. Silberhorn, E. Konig, P.K. Lam, A. Sizmann, N. Korolkova. Index.

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Quantum Information with Continuous Variables

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

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Gaussian quantum information

An overview of the recent developments in the field of cylindrical vector beams is
provided. As one class of spatially variant polarization, cylindrical vector beams
are the axially symmetric beam solution to the full vector electromagnetic wave
equation. These beams can be generated via different active and passive methods.
Techniques for manipulating these beams while maintaining the polarization symmetry
have also been developed. Their special polarization symmetry gives rise to unique
high-numerical-aperture focusing properties that find important applications in
nanoscale optical imaging and manipulation. The prospects for cylindrical vector
beams and their applications in other fields are also briefly discussed.

Cylindrical vector beams: from mathematical concepts to applications

Quantum teleportation of optical coherent states was demonstrated experimentally using squeezed-state entanglement. The quantum nature of the achieved teleportation was verified by the experimentally determined fidelity Fexp = 0.58 +/- 0.02, which describes the match between input and output states. A fidelity greater than 0.5 is not possible for coherent states without the use of entanglement. This is the first realization of unconditional quantum teleportation where every state entering the device is actually teleported.

http://science.sciencemag.org/content/sci/282/5389/706.full.pdf

Unconditional quantum teleportation

The first quantum technology that harnesses quantum mechanical effects for its core operation has arrived in the form of commercially available quantum key distribution systems. This technology achieves enhanced security by encoding information in photons such that an eavesdropper in the system can be detected. Anticipated future quantum technologies include large-scale secure networks, enhanced measurement and lithography, and quantum information processors, which promise exponentially greater computational power for particular tasks. Photonics is destined to have a central role in such technologies owing to the high-speed transmission and outstanding low-noise properties of photons. These technologies may use single photons, quantum states of bright laser beams or both, and will undoubtedly apply and drive state-of-the-art developments in photonics.

Photonic quantum technologies

The Einstein-Podolsky-Rosen paradox is demonstrated experimentally for dynamical variables having a continuous spectrum. As opposed to previous work with discrete spin or polarization variables, the continuous optical amplitudes of a signal beam are inferred in turn from those of a spatially separated but strongly correlated idler beam generated by nondegenerate parametric amplification. The uncertainty product for the variances of these inferences is observed to be 0.70±0.01, which is below the limit of unity required for the demonstration of the paradox.

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Realization of the Einstein-Podolsky-Rosen paradox for continuous variables.

We report the reconstruction of the quantum state of squeezed vacuum generated by a continuous-wave optical parametric amplifier. Homodyne detection and tomographic reconstruction methods were used to obtain the density matrix in the Fock (number state) representation. The photon number distribution exhibits odd-even oscillations, a manifestation of the photon pair production in the second-order nonlinear medium.

http://www.exphy.uni-duesseldorf.de/Publikationen/1996/prl77_2933.pdf

Quantum Statistics of the Squeezed Vacuum by Measurement of the Density Matrix in the Number State Representation.

Squeezed states of light are generated by the process of second-harmonic conversion within an optical cavity resonant at both fundamental and harmonic frequencies. Observations of squeezing are made by analyzing the spectral density of photocurrent fluctuations produced by the total field reflected from the nonlinear cavity. Reductions in photocurrent noise of 13% relative to the coherent-state or shot-noise level are achieved for frequency offsets near 4 MHz.

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Generation of squeezed light by intracavity frequency doubling.

Conversion efficiency of 85% has been achieved in cw second-harmonic generation from 1.08 to 0.54 μm with a potassium titanyl phosphate crystal inside an external ring cavity. An absolute comparison between the experimental data and a simple theory is made and shows good agreement.

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85% efficiency for cw frequency doubling from 1.08 to 0.54 μm

The Einstein-Podolsky-Rosen (EPR) paradox is demonstrated experimentally for continuous variables by employing a nondegenerate optical parametric amplifier (NOPA). Such a system is analogous to and under some ideal conditions is in one-to-one correspondence with the original system discussed by EPR. In particular, the quadrature-phase amplitudes for a signal beam are inferred in turn from those of a spatially separated but strongly correlated idler beam, where these optical amplitudes are analogous to canonical position and momentum variables. The variances for the two inferences are measured and their product is observed to be below the limit of unity associated with the Heisenberg uncertainty relation, in apparent contradiction with quantum mechanics according to the argument of EPR. The smallest product of inference variances achieved in the experiment is (0.70±0.01). Various other types of quantum noise for this system are also investigated, and a theory of a narrowband NOPA is presented with losses included. A comparison between experiment and this theory shows relatively good agreement. The question of a local hidden-variables description of the system is discussed.

Silvania F. Pereira

Papers

Realization of the Einstein-Podolsky-Rosen paradox for continuous variables.

Quantum Statistics of the Squeezed Vacuum by Measurement of the Density Matrix in the Number State Representation.

Generation of squeezed light by intracavity frequency doubling.

85% efficiency for cw frequency doubling from 1.08 to 0.54 μm

Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification